Higher interactions and additive renormalization

We should stress finally that the arguments presented in this chapter are important fcjr a deeper understanding of our model, but are no prerequisite for following the main thread of this treatise. 

The inequality (10.2) shows that the estimate (10,1) doe.s not count the number of -segment contacts correctly. To save the estimate wc can interpret it as number of contacts among segments which are spaced along the chain by somti finite fraction of the total cliaiii length. Indeed, evaluating, which is properly defintid as 

What is the effect of interactions within small fractions of the chain Implicitly we have mswc. nid this qinistion before, in the ciilc ulation of (Sect, 5.4.3), for instance. Wo just have to recollect some re sults. We consider the free energy of a single chain, defined as 

The leading contribution n is dominated by the summation over small fractions j2 — ji otn of the chain. It can be interpreted as a correction to the segment chemical potential ps  

Now such terms also result from one-body potentials. Recall the general decomposition (2.9) of the effective segment potential  

As mentioned above, the matter is quite involved and we will only sketch the arguments. Discussing some low order terms we will demonstrate the additive renormalizations due to higher order interactions (Sect. 10.1). Then we will discuss the general ideas on the structure of the renormalization group, defining important concepts like relevance or irrelevance of interactions or the critical manifold (Sect. 10.2). Concerning the field theoretic realization of the RG, we will summarize some results (Sect. 10.3). 

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This redefinition is another example of additive renormalization. In principle it can be implemented perturbatively to all orders. As a result, not the mathematics of our model but the interpretation of changes. 0K sums up contributions from all higher interactions and thus should be viewed as a parameterization of the interaction among whole strands of the chains rather than as a true two-segrnent interaction. This also makes clear that there is no hope to calculate 0e from a realistic model. 

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